Integrand size = 25, antiderivative size = 124 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=-\frac {7 c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}+\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2} \]
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Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6313, 893, 887, 889, 214} \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2}+\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 \sqrt {c-\frac {c}{a x}}}-\frac {7 c x \sqrt {1-\frac {1}{a^2 x^2}}}{4 a \sqrt {c-\frac {c}{a x}}} \]
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Rule 214
Rule 887
Rule 889
Rule 893
Rule 6313
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}+\frac {7 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{4 a} \\ & = -\frac {7 c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}-\frac {7 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2} \\ & = -\frac {7 c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}-\frac {\left (7 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^4} \\ & = -\frac {7 c \sqrt {1-\frac {1}{a^2 x^2}} x}{4 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{2 \sqrt {c-\frac {c}{a x}}}+\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{4 a^2} \\ \end{align*}
Time = 1.49 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.12 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 (-7+2 a x)}{-4+4 a x}-\frac {7 \sqrt {c} \log (1-a x)}{8 a^2}+\frac {7 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{8 a^2} \]
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Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-14 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+7 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{8 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}}\) | \(116\) |
risch | \(\frac {\left (2 a x -7\right ) x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{4 a \left (a x -1\right )}+\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{8 a \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(152\) |
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Time = 0.28 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.59 \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\left [\frac {7 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (2 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} x - a^{2}\right )}}, -\frac {7 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (2 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} x - a^{2}\right )}}\right ] \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int x \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}\, dx \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int { \sqrt {c - \frac {c}{a x}} x \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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\[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int { \sqrt {c - \frac {c}{a x}} x \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int x\,\sqrt {c-\frac {c}{a\,x}}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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